![Manifolds
Hilbert Spaces and Kernels
Kernels on Manifolds
Applications and Experiments
Kernels on Sym+
d
Number of different metrics have been proposed for Sym+
d to
capture its nonlinearity.
Only some of them are true geodesic distances that arise from
Riemannian metrics.
Metric on Sym+
d
Geodesic Distance
Positive Definite
Gaussian ∀σ > 0
Log-Euclidean Yes Yes
Affine-Invariant Yes No
Cholesky No Yes
Power-Euclidean No Yes
Root Stein Divergence No No*
Log-Euclidean e− log(S1)−log(S2) 2/2σ2
Affine-Invariant e− log(S
−1/2
1 S2S
−1/2
1 ) 2/2σ2
Root Stein Divergence [det(S1) det(S2)/det(S1/2 + S2/2)]
1
2σ2
Sadeep Jayasumana, Richard Hartley, Mathieu Salzmann, Hongdong Li, Mehrtash HarandiKernel Methods on the Riemannian Manifold of Symmetric Positi](https://image.slidesharecdn.com/cvpr2013presentation-160224141410/85/Kernel-Methods-on-Manifolds-15-320.jpg)
Kernel Methods on Manifolds
- 1. Manifolds Hilbert Spaces and Kernels Kernels on Manifolds Applications and Experiments Kernel Methods on the Riemannian Manifold of Symmetric Positive Definite Matrices Sadeep Jayasumana, Richard Hartley, Mathieu Salzmann, Hongdong Li, Mehrtash Harandi June 25, 2013 Sadeep Jayasumana, Richard Hartley, Mathieu Salzmann, Hongdong Li, Mehrtash HarandiKernel Methods on the Riemannian Manifold of Symmetric Positi
- 2. Manifolds Hilbert Spaces and Kernels Kernels on Manifolds Applications and Experiments Contents 1 Manifolds 2 Hilbert Spaces and Kernels 3 Kernels on Manifolds 4 Applications and Experiments Sadeep Jayasumana, Richard Hartley, Mathieu Salzmann, Hongdong Li, Mehrtash HarandiKernel Methods on the Riemannian Manifold of Symmetric Positi
- 3. Manifolds Hilbert Spaces and Kernels Kernels on Manifolds Applications and Experiments What is a manifold? Topological Manifolds Think of a (topological) manifold as a smooth surface in Rn. Every point on the manifold has a neighbourhood that is the same as (homeomorphic to) an open ball in Rn. Differentiable Manifolds A manifold with a differentiable structure. Riemannian Manifolds A differentiable manifold with a family smoothly varying inner products (Riemannian metric) on tangent spaces. Sadeep Jayasumana, Richard Hartley, Mathieu Salzmann, Hongdong Li, Mehrtash HarandiKernel Methods on the Riemannian Manifold of Symmetric Positi
- 4. Manifolds Hilbert Spaces and Kernels Kernels on Manifolds Applications and Experiments Examples of manifolds Rn. Sphere Sn. Rotation space SO(3). Grassmann manifolds – used to model sets of images. Essential manifold – structure and motion. Shape manifolds – capture the shape of an object. Symmetric Positive Definite (SPD) matrices – covariance features, diffusion tensors, structure tensors. Sadeep Jayasumana, Richard Hartley, Mathieu Salzmann, Hongdong Li, Mehrtash HarandiKernel Methods on the Riemannian Manifold of Symmetric Positi
- 5. Manifolds Hilbert Spaces and Kernels Kernels on Manifolds Applications and Experiments Terms An Inner Product Space is a vector space with an inner product defined on it. A Hilbert Space is a complete inner product space. Usually, when you talk of a Hilbert space, you think of it being infinite dimensional. A norm defines the length of vectors in a vector space. A norm allows you to measure distances; an inner product naturally induces a norm. d(x, y) = x − y = x − y, x − y A Metric Space is a set (not necessarily a vector space) with the distance between its elements (metric) defined. Sadeep Jayasumana, Richard Hartley, Mathieu Salzmann, Hongdong Li, Mehrtash HarandiKernel Methods on the Riemannian Manifold of Symmetric Positi
- 6. Manifolds Hilbert Spaces and Kernels Kernels on Manifolds Applications and Experiments Reproducing kernels Let X be a set. A Reproducing Kernel is a real (resp. complex)-valued function on X × X such that, it defines the inner product of a Hilbert space of functions from X to R (resp. C). k(x, y) = fx , fy H where fx , fy ∈ RX According to Mercer’s theorem, only positive definite kernels define with Reproducing Kernel Hilbert Spaces. This theory is extremely useful since reproducing kernels can be utilized to evalutate inner products in infinite dimensional spaces of functions. Sadeep Jayasumana, Richard Hartley, Mathieu Salzmann, Hongdong Li, Mehrtash HarandiKernel Methods on the Riemannian Manifold of Symmetric Positi
- 7. Manifolds Hilbert Spaces and Kernels Kernels on Manifolds Applications and Experiments Why kernels on manifolds? Lets us perform well known algorithms designed for linear spaces, on nonlinear manifolds. Support vector machines. Principal component analysis. Discriminant analysis. Multiple kernel learning. k-means. Embedding a manifold in a Hilbert space yields a much richer data representation as we are mapping from a low dimensional space to a high dimensional space. Aids finding non-trivial patterns. Think about linear SVM vs kernel SVM. Sadeep Jayasumana, Richard Hartley, Mathieu Salzmann, Hongdong Li, Mehrtash HarandiKernel Methods on the Riemannian Manifold of Symmetric Positi
- 8. Manifolds Hilbert Spaces and Kernels Kernels on Manifolds Applications and Experiments Defining kernels on manifolds Gaussian RBF, the most commonly used kernel in Rn. kG (x, y) = e− x−y 2/σ2 This kernel is always positive definite for all σ. Sadeep Jayasumana, Richard Hartley, Mathieu Salzmann, Hongdong Li, Mehrtash HarandiKernel Methods on the Riemannian Manifold of Symmetric Positi
- 9. Manifolds Hilbert Spaces and Kernels Kernels on Manifolds Applications and Experiments Defining kernels on manifolds Gaussian RBF, the most commonly used kernel in Rn. kG (x, y) = e− x−y 2/σ2 This kernel is always positive definite for all σ. Replace the Euclidean distance with the geodesic distance on the Riemannian manifold? kR(x, y) = e−d2(x,y)/σ2 Sadeep Jayasumana, Richard Hartley, Mathieu Salzmann, Hongdong Li, Mehrtash HarandiKernel Methods on the Riemannian Manifold of Symmetric Positi
- 10. Manifolds Hilbert Spaces and Kernels Kernels on Manifolds Applications and Experiments Defining kernels on manifolds Gaussian RBF, the most commonly used kernel in Rn. kG (x, y) = e− x−y 2/σ2 This kernel is always positive definite for all σ. Replace the Euclidean distance with the geodesic distance on the Riemannian manifold? kR(x, y) = e−d2(x,y)/σ2 Does NOT give a positive definite kernel always! Eg. Geodesic (great-circle) distance on S2 , Affine-invariant geodesic distance on Sym+ d . Sadeep Jayasumana, Richard Hartley, Mathieu Salzmann, Hongdong Li, Mehrtash HarandiKernel Methods on the Riemannian Manifold of Symmetric Positi
- 11. Manifolds Hilbert Spaces and Kernels Kernels on Manifolds Applications and Experiments Defining kernels on manifolds Gaussian RBF, the most commonly used kernel in Rn. kG (x, y) = e− x−y 2/σ2 This kernel is always positive definite for all σ. Replace the Euclidean distance with the geodesic distance on the Riemannian manifold? kR(x, y) = e−d2(x,y)/σ2 Does NOT give a positive definite kernel always! Eg. Geodesic (great-circle) distance on S2 , Affine-invariant geodesic distance on Sym+ d . Under what conditions is this positive definite? Sadeep Jayasumana, Richard Hartley, Mathieu Salzmann, Hongdong Li, Mehrtash HarandiKernel Methods on the Riemannian Manifold of Symmetric Positi
- 12. Manifolds Hilbert Spaces and Kernels Kernels on Manifolds Applications and Experiments The Gaussian RBF on a metric space The Gaussian RBF on a metric space (M, d) is positive definite for all σ > 0 if and only if d2 is negative definite. Sadeep Jayasumana, Richard Hartley, Mathieu Salzmann, Hongdong Li, Mehrtash HarandiKernel Methods on the Riemannian Manifold of Symmetric Positi
- 13. Manifolds Hilbert Spaces and Kernels Kernels on Manifolds Applications and Experiments The Gaussian RBF on a metric space The Gaussian RBF on a metric space (M, d) is positive definite for all σ > 0 if and only if d2 is negative definite. d2 is negative definite if and only if (M, d) is isometrically embeddable in a Hilbert space. Sadeep Jayasumana, Richard Hartley, Mathieu Salzmann, Hongdong Li, Mehrtash HarandiKernel Methods on the Riemannian Manifold of Symmetric Positi
- 14. Manifolds Hilbert Spaces and Kernels Kernels on Manifolds Applications and Experiments The Gaussian RBF on a metric space The Gaussian RBF on a metric space (M, d) is positive definite for all σ > 0 if and only if d2 is negative definite. d2 is negative definite if and only if (M, d) is isometrically embeddable in a Hilbert space. Theorem : Gaussian RBF on metric spaces Let (M, d) be a metric space and define k : (M × M) → R by k(xi , xj ) := exp(−d2(xi , xj )/2σ2). Then, k is a positive definite kernel for all σ > 0 if and only if there exists an inner product space V and a function ψ : M → V such that, d(xi , xj ) = ψ(xi ) − ψ(xj ) V. Sadeep Jayasumana, Richard Hartley, Mathieu Salzmann, Hongdong Li, Mehrtash HarandiKernel Methods on the Riemannian Manifold of Symmetric Positi
- 15. Manifolds Hilbert Spaces and Kernels Kernels on Manifolds Applications and Experiments Kernels on Sym+ d Number of different metrics have been proposed for Sym+ d to capture its nonlinearity. Only some of them are true geodesic distances that arise from Riemannian metrics. Metric on Sym+ d Geodesic Distance Positive Definite Gaussian ∀σ > 0 Log-Euclidean Yes Yes Affine-Invariant Yes No Cholesky No Yes Power-Euclidean No Yes Root Stein Divergence No No* Log-Euclidean e− log(S1)−log(S2) 2/2σ2 Affine-Invariant e− log(S −1/2 1 S2S −1/2 1 ) 2/2σ2 Root Stein Divergence [det(S1) det(S2)/det(S1/2 + S2/2)] 1 2σ2 Sadeep Jayasumana, Richard Hartley, Mathieu Salzmann, Hongdong Li, Mehrtash HarandiKernel Methods on the Riemannian Manifold of Symmetric Positi
- 16. Manifolds Hilbert Spaces and Kernels Kernels on Manifolds Applications and Experiments Pedestrian detection Table: Sample images from INRIA dataset Sadeep Jayasumana, Richard Hartley, Mathieu Salzmann, Hongdong Li, Mehrtash HarandiKernel Methods on the Riemannian Manifold of Symmetric Positi
- 17. Manifolds Hilbert Spaces and Kernels Kernels on Manifolds Applications and Experiments Pedestrian detection Covariance descriptor is used as the region descriptor following Tuzel et al., 2008. Multiple covariance descriptors are calculated per detection window, an SVM + MKL framework is used to build the classifier. 10 −5 10 −4 10 −3 10 −2 10 −1 0.01 0.02 0.05 0.1 0.2 0.3 False Positives Per Window (FPPW) MissRate Proposed Method (MKL on Manifold) MKL with Euclidean Kernel LogitBoost on Manifold HOG Kernel SVM HOG Linear SVM Sadeep Jayasumana, Richard Hartley, Mathieu Salzmann, Hongdong Li, Mehrtash HarandiKernel Methods on the Riemannian Manifold of Symmetric Positi
- 18. Manifolds Hilbert Spaces and Kernels Kernels on Manifolds Applications and Experiments Visual object categorization Table: Sample images from ETH-80 dataset Sadeep Jayasumana, Richard Hartley, Mathieu Salzmann, Hongdong Li, Mehrtash HarandiKernel Methods on the Riemannian Manifold of Symmetric Positi
- 19. Manifolds Hilbert Spaces and Kernels Kernels on Manifolds Applications and Experiments Visual object categorization ETH-80 dataset, 5 × 5 covariance descriptors. Manifold k-means and manifold kernel k-means with different metrics. Nb. of Euclidean Cholesky Power-Euclidean Log-Euclidean classes KM KKM KM KKM KM KKM KM KKM 3 72.50 79.00 73.17 82.67 71.33 84.33 75.00 94.83 4 64.88 73.75 69.50 84.62 69.50 83.50 73.00 87.50 5 54.80 70.30 70.80 82.40 70.20 82.40 74.60 85.90 6 50.42 69.00 59.83 73.58 59.42 73.17 66.50 74.50 7 42.57 68.86 50.36 69.79 50.14 69.71 59.64 73.14 8 40.19 68.00 53.81 69.44 54.62 68.44 58.31 71.44 Sadeep Jayasumana, Richard Hartley, Mathieu Salzmann, Hongdong Li, Mehrtash HarandiKernel Methods on the Riemannian Manifold of Symmetric Positi
- 20. Manifolds Hilbert Spaces and Kernels Kernels on Manifolds Applications and Experiments Texture recognition Table: Sample images from Brodatz Sadeep Jayasumana, Richard Hartley, Mathieu Salzmann, Hongdong Li, Mehrtash HarandiKernel Methods on the Riemannian Manifold of Symmetric Positi
- 21. Manifolds Hilbert Spaces and Kernels Kernels on Manifolds Applications and Experiments Texture recognition Demontrating kernel PCA on manifolds with the proposed kernel. The proposed kernel achieves better results than the usual Euclidean kernel that neglects the Riemannian geometry of the space. Kernel Classification Accuracy l = 10 l = 11 l = 12 l = 15 Riemannian 95.50 95.95 96.40 96.40 Euclidean 89.64 90.09 90.99 91.89 Table: Texture recognition. Recognition accuracies on the Brodatz dataset with k-NN in a l-dimensional Euclidean space obtained by kernel PCA. Sadeep Jayasumana, Richard Hartley, Mathieu Salzmann, Hongdong Li, Mehrtash HarandiKernel Methods on the Riemannian Manifold of Symmetric Positi
- 22. Manifolds Hilbert Spaces and Kernels Kernels on Manifolds Applications and Experiments DTI segmentation Diffusion tensor at the voxel is directly used as the descriptor. Kernel k-means is utilized to cluster points on Sym+ d , yielding a segmentation of the DTI image. Ellipsoids Fractional Anisotropy Riemannian kernel Euclidean kernel Sadeep Jayasumana, Richard Hartley, Mathieu Salzmann, Hongdong Li, Mehrtash HarandiKernel Methods on the Riemannian Manifold of Symmetric Positi
- 23. Manifolds Hilbert Spaces and Kernels Kernels on Manifolds Applications and Experiments Motion segmentation The structure tensor (3 × 3) was used as the descriptor. Kernel k-means clustering of the tensors yields the segmentation. Achieves better clustering accuracy than methods that work in a low dimensional space. Frame 1 Frame 2 KKM on Sym+ 3 LLE on Sym+ 3 LE on Sym+ 3 HLLE on Sym+ 3 Sadeep Jayasumana, Richard Hartley, Mathieu Salzmann, Hongdong Li, Mehrtash HarandiKernel Methods on the Riemannian Manifold of Symmetric Positi
- 24. Manifolds Hilbert Spaces and Kernels Kernels on Manifolds Applications and Experiments Questions? Sadeep Jayasumana, Richard Hartley, Mathieu Salzmann, Hongdong Li, Mehrtash HarandiKernel Methods on the Riemannian Manifold of Symmetric Positi